Merge branch 'AndreasFMueller:master' into master

10 files changed, 466 insertions, 0 deletions

diff --git a/vorlesungen/slides/dreieck/Makefile.inc b/vorlesungen/slides/dreieck/Makefile.inc
new file mode 100644
index 0000000..bbc19b6
--- /dev/null
+++ b/vorlesungen/slides/dreieck/Makefile.inc
@@ -0,0 +1,14 @@
+#
+# Makefile.inc 
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapterdreieck =							\
+	../slides/dreieck/stichprobe.tex				\
+	../slides/dreieck/minmax.tex					\
+	../slides/dreieck/ordnungsstatistik.tex				\
+	../slides/dreieck/orderplot.tex					\
+	../slides/dreieck/dichte.tex					\
+	../slides/dreieck/beta.tex					\
+	../slides/dreieck/betaplot.tex					\
+	../slides/dreieck/test.tex
diff --git a/vorlesungen/slides/dreieck/beta.tex b/vorlesungen/slides/dreieck/beta.tex
new file mode 100644
index 0000000..fc3606a
--- /dev/null
+++ b/vorlesungen/slides/dreieck/beta.tex
@@ -0,0 +1,70 @@
+%
+% beta.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beta-Verteilung}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{block}{Ordnungsstatistik}
+\begin{align*}
+\varphi(x)
+&=
+{\color{blue}N} x^{k-1} (1-x)^{n-k}
+\\
+&\uncover<8->{
+=
+\beta_{k,n-k+1}(x)
+}
+\end{align*}
+\end{block}
+\uncover<8->{%
+\begin{block}{Risch-Algorithmus}
+Die Beta-Verteilungen haben ausser in Spezialfällen
+keine Stammfunktion in geschlossener Form.
+\end{block}}
+\end{column}
+\begin{column}{0.56\textwidth}
+\uncover<2->{%
+\begin{definition}
+Beta-Verteilung
+\[
+\beta_{a,b}(x)
+=
+\begin{cases}
+\displaystyle
+\uncover<7->{
+{\color{blue}
+\frac{1}{B(a,b)}
+}
+}
+x^{a-1}(1-x)^{b-1}
+&0\le x\le 1
+\\
+0&\text{sonst}
+\end{cases}
+\]
+\end{definition}}
+\uncover<3->{%
+\begin{block}{Normierung}
+\begin{align*}
+{\color{blue}\frac{1}{{N}}}
+&\uncover<4->{=
+\int_{-\infty}^\infty \beta_{a,b}(x)\,dx}
+\\
+&\uncover<5->{=
+\int_{0}^1 x^{a-1}(1-x)^{b-1}\,dx}
+\\
+&\uncover<6->{=
+B(a,b)}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/dreieck/betaplot.tex b/vorlesungen/slides/dreieck/betaplot.tex
new file mode 100644
index 0000000..ee932e8
--- /dev/null
+++ b/vorlesungen/slides/dreieck/betaplot.tex
@@ -0,0 +1,38 @@
+%
+% betaplot.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Beta-Verteilungen}
+\begin{center}
+\begin{tikzpicture}[>=latex]
+
+\only<1>{
+\begin{scope}
+	\clip (-7,-3.2) rectangle (7,3.2);
+	\node at (0,-6.5) {\includegraphics[width=13.5cm]{../../buch/chapters/040-rekursion/images/beta.pdf}};
+\end{scope}
+}
+
+\only<2>{
+\begin{scope}
+	\clip (-7,-3.2) rectangle (7,3.2);
+	\node at (0,-0) {\includegraphics[width=13.5cm]{../../buch/chapters/040-rekursion/images/beta.pdf}};
+\end{scope}
+}
+
+\only<3>{
+\begin{scope}
+	\clip (-7,-3.2) rectangle (7,3.2);
+	\node at (0,6.5) {\includegraphics[width=13.5cm]{../../buch/chapters/040-rekursion/images/beta.pdf}};
+\end{scope}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/dreieck/chapter.tex b/vorlesungen/slides/dreieck/chapter.tex
new file mode 100644
index 0000000..0f58c4c
--- /dev/null
+++ b/vorlesungen/slides/dreieck/chapter.tex
@@ -0,0 +1,11 @@
+%
+% chapter.tex -- slides for chapter dreieck
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\folie{dreieck/test.tex}
+\folie{dreieck/minmax.tex}
+\folie{dreieck/ordnungsstatistik.tex}
+\folie{dreieck/dichte.tex}
+\folie{dreieck/beta.tex}
+\folie{dreieck/betaplot.tex}
diff --git a/vorlesungen/slides/dreieck/dichte.tex b/vorlesungen/slides/dreieck/dichte.tex
new file mode 100644
index 0000000..168523a
--- /dev/null
+++ b/vorlesungen/slides/dreieck/dichte.tex
@@ -0,0 +1,67 @@
+%
+% dichte.tex -- Wahrscheinlichkeitsdichte
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Wahrscheinlichkeitsdichte}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{block}{Definition}
+\[
+\varphi_{X_{k:n}}(x)
+=
+\frac{d}{dx} F_{X_{k:n}}(x)
+\]
+\end{block}
+\end{column}
+\begin{column}{0.60\textwidth}
+\uncover<4->{%
+\begin{block}{Gleichverteilung}
+\[
+{\color{darkgreen}F(x)}=\begin{cases}
+0&x \le 0\\
+x&0\le x \le 1,\\
+1&x\ge 1
+\end{cases}
+\quad
+\uncover<5->{
+{\color{red}\varphi(x)}
+=
+\begin{cases}
+1&0\le x \le 1\\
+0&\text{sonst}
+\end{cases}}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<2->{%
+\begin{block}{Ordnungsstatistik}
+nach einiger Rechnung:
+\begin{align*}
+\varphi_{X_{k:n}}(x)
+&=
+{\color<3->{red}\varphi_X(x)}\,k\binom{n}{k}{\color<3->{darkgreen}F_X(x)}^{k-1}
+(1-{\color<3->{darkgreen}F_X(x)})^{n-k}
+\intertext{\uncover<4->{für Gleichverteilung}}
+\uncover<6->{
+\varphi_{X_{k:n}}(x)
+&=
+\begin{cases}
+\displaystyle
+{\color<7->{blue}k\binom{n}{k}}{\color{darkgreen}x}^{k-1}(1-{\color{darkgreen}x})^{n-k}
+&0\le x \le 1\\
+0&\text{sonst}
+\end{cases}
+\qquad\uncover<7->{\text{({\color{blue}Normierung})}}
+}
+\end{align*}
+\end{block}}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/dreieck/minmax.tex b/vorlesungen/slides/dreieck/minmax.tex
new file mode 100644
index 0000000..ff3a231
--- /dev/null
+++ b/vorlesungen/slides/dreieck/minmax.tex
@@ -0,0 +1,83 @@
+%
+% minmax.tex -- Minimum und Maximum
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Minimum und Maximum}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Maximum}
+Verteilungsfunktion von
+\[
+Z=\operatorname{max}(X_1,\dots,X_n)
+\]
+\begin{align*}
+\uncover<3->{
+F_Z(x)
+&=
+P(Z\le x)}
+\\
+\uncover<4->{
+&=
+P(X_1\le x\wedge\dots\wedge X_n\le x)
+}
+\\
+\uncover<5->{
+&=
+P(X_1\le x)\cdot \ldots\cdot P(X_n\le x)
+}
+\\
+\uncover<6->{
+&=
+F_X(x)^n
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Minimum}
+Verteilungsfunktion von
+\[
+Z=\operatorname{min}(X_1,\dots,X_n)
+\]
+\begin{align*}
+\uncover<7->{
+F_Z(x)
+&=
+P(Z\le x)
+}
+\\
+\uncover<8->{
+&=P(\overline{
+X_1\le x\wedge\dots\wedge X_n \le x
+})
+}
+\\
+\uncover<9->{
+&=
+1-P(
+X_1> x\wedge\dots\wedge X_n > x
+)
+}
+\\
+\uncover<10->{
+&=
+1-(P(X_1>x)\cdot\ldots\cdot P(X_n>x))
+}
+\\
+\uncover<11->{
+&=
+1-(1-F_X(x))^n
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/dreieck/orderplot.tex b/vorlesungen/slides/dreieck/orderplot.tex
new file mode 100644
index 0000000..7cf10c6
--- /dev/null
+++ b/vorlesungen/slides/dreieck/orderplot.tex
@@ -0,0 +1,16 @@
+%
+% orderplot.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ordnungstatistik}
+\vspace*{-18pt}
+\begin{center}
+\includegraphics[width=10cm]{../../buch/chapters/040-rekursion/images/order.pdf}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/dreieck/ordnungsstatistik.tex b/vorlesungen/slides/dreieck/ordnungsstatistik.tex
new file mode 100644
index 0000000..c968e79
--- /dev/null
+++ b/vorlesungen/slides/dreieck/ordnungsstatistik.tex
@@ -0,0 +1,84 @@
+%
+% ordnungsstatistik.tex -- Ordnungsstatistik
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ordnungstatistik}
+\vspace{-10pt}
+\begin{block}{Angeordnete Stichprobe}
+\[
+X_{1:n}
+\le
+X_{2:n}
+\le
+\dots
+\le
+X_{(n-1):n}
+\le
+X_{n:n}
+\]
+$X_{k:n} = \mathstrut$der $k$-te von $n$ Werten
+\end{block}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.44\textwidth}
+\uncover<2->{%
+\begin{block}{Verteilungsfunktion}
+\begin{align*}
+F_{X_{k:n}}(x)
+&=
+P(X_{k:n} \le x)
+\\
+&\uncover<3->{=
+P\bigl(
+|\{i\;|\; {\color<4>{red}X_i\le x}\}| \ge k
+\bigr)}
+\\
+&\uncover<5->{=
+P(\text{Anzahl $A_i$}\ge k)}
+\\
+&\uncover<9->{=
+P(K\ge k)}
+\\
+\uncover<6->{
+F_{X_i}(x)&= P(X_i\le x)}\uncover<7->{ = P(A_i)}\uncover<10->{ = p}
+}
+\end{align*}
+\uncover<4->{$A_i=\{X_i\le x\}$}\uncover<7->{ ist ein Beroulli- Experiment 
+\uncover<10->{mit Eintretens- wahrscheinlichkeit $p$}
+\end{block}}
+\end{column}
+\begin{column}{0.52\textwidth}
+\uncover<8->{%
+\begin{block}{Wiederholtes Bernoulli-Experiment}
+$K=\mathstrut$Anzahl $k$, für die $A$ eingetreten
+ist\only<11->{, ist binomialverteilt:}
+\begin{align*}
+\uncover<12->{P(K=k) 
+&=
+\phantom{\sum_{i=k}^n\mathstrut}
+\binom{n}{k} p^k (1-p)^{n-k}
+}
+\\
+\uncover<13->{
+P(K\ge k)
+&=
+\sum_{i=k}^n
+\binom{n}{i} p^i (1-p)^{n-i}
+}
+\\
+\uncover<14->{
+&=
+\sum_{i=k}^n
+\binom{n}{i} F_X(x)^i (1-F_X(x))^{n-i}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/dreieck/stichprobe.tex b/vorlesungen/slides/dreieck/stichprobe.tex
new file mode 100644
index 0000000..4b2eff0
--- /dev/null
+++ b/vorlesungen/slides/dreieck/stichprobe.tex
@@ -0,0 +1,64 @@
+%
+% stichprobe.tex -- Stichprobe
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Stichprobe}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Zufallsvariable}
+Gegeben eine Zufallsvariable $X$ \uncover<5->{mit
+Verteilungsfunktion
+\[
+F_X(x)
+=
+P(X\le x)
+\]}
+\uncover<6->{und
+Wahrscheinlichkeitsdichte
+\[
+\varphi_X(x)
+=
+\frac{d}{dx} F_X(x)
+\]}
+\end{block}
+\uncover<7->{%
+\begin{block}{Gleichverteilung}
+\[
+F(x) = \begin{cases}
+0&\qquad x<0\\
+x&\qquad 0\le x \le 1\\
+1&\qquad 1<x
+\end{cases}
+\uncover<8->{
+\qquad\Rightarrow\qquad
+\varphi(x)
+=
+\begin{cases}
+1&\qquad 0\le x \le 1\\
+0&\qquad\text{sonst}.
+\end{cases}
+}
+\]
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Stichprobe}
+$n$ Zufallsvariablen $X_1,\dots,X_n$
+\begin{itemize}
+\item<3->
+alle $X_i$ haben die gleiche Verteilung wie $X$
+\item<4->
+die $X_i$ sind unabhängig
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/dreieck/test.tex b/vorlesungen/slides/dreieck/test.tex
new file mode 100644
index 0000000..117d03d
--- /dev/null
+++ b/vorlesungen/slides/dreieck/test.tex
@@ -0,0 +1,19 @@
+%
+% template.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Template für Dreieckstest}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\end{column}
+\begin{column}{0.48\textwidth}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup